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5.80 Small-Molecule Spectroscopy and Dynamics
Fall 2008
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5.80 Lecture #19
Fall, 2008
Page 1 of 8 pages
Lecture #19: Second-Order Effects
Last time:
perturbations = accidental degeneracy
Today:
effects of “Remote Perturbers”. What terms must we add to the effective H so that we
can represent all usual behaviors with minimum number of parameters.
Use the van Vleck transformation.
Two effects to be discussed
*
centrifugal distortion of all zero- and first-order parameters.
e.g.
B→D
[explicit R-dependence of B(R)]
A → AD
[implicit R-dependence of A(R)]
[interaction with all v’s of same Λ-S state]
*
Λ-doubling and other 2nd-order parameters [interaction with all v’s of all other states]
We will work with 2∏, 2∑s example
Recipe
*
Heff in terms of E, B, A, (λ, γ), α, β
*
van Vleck transformation: diagrammatically in the form of “railroads” for each location
in Heff
*
each term in van Vleck transformation is
explicit function
f ( v, J ) *
∑
e ′, v′
H ev,e′v′ H e′v ′,ev
o
E oev − E e′
v′
new 2nd order
parameter
2
2
e/f
∏3/2
∏1/2
2
∑s
2
∏3/2
E v∏ + A∏ 2 + Bv∏ ( y 2 − 2 )
2
2
∏1/2
−Bv∏ ( y 2 −1)
1/2
E v∏ − A∏ 2 + Bv∏ ( y 2 )
∑s
−βsv∏ v∑ ( y 2 −1)
1/2
αs + βs ⎡⎣1  (−1)s y⎤⎦
E v∑ + Bv∑ ⎡⎣y 2  (−1)s y⎤⎦
y ≡ J + 1/2
For simplicity we do not include γ terms (λ terms are not possible for S < 1 states).
5.80 Lecture #19
Fall, 2008
Page 2 of 8 pages
What do we do with these?
⎛
m
interesting
⎞
H mn
′ ⎟
⎜
m′
⎟
⎜
H =
⎟
⎜ n
remote
⎜
⎟
⎝
n′
⎠
follows rules for
matrix
multiplication
VV
H m,m′
≡ E om δ mm′ + λ1 H m′ m′
λ2
+
2
∑
n
′ H n′ m′ H mn
′ H n′ m′ ⎤
⎡ H mn
+
⎢ Eo − Eo
E om′ − E on ⎥⎦
n
⎣ m
~ λ2
∑
n
H′mn H′nm′
E om + E om ′
− E on
2
We are going to write Heff in terms of
zero-order parameters
E, B, A
perturbation parameters
α, β
second-order parameters
D, AD, o, p, q
 =H
 ROT + H
 SO
H
)
(H
2
(
 ROT
= H
(
 SO
+ H
(
)
)
2
2
 ROT ⊗ H
 SO
+ H
)2
e/f dependent
e/f independent
q (Λ-doubling)
D (centrifugal distortion of B)
e/f dependent
e/f independent
o (Λ-doubling)
λ (2nd-order spin-spin)
e/f dependent
e/f independent
p (Λ-doubling)
γ (2nd-order spin-rotation)
AD (centrifugal distortion of A)
Generate many 2nd-order parameters — not all are linearly independent.
5.80 Lecture #19
Fall, 2008
Page 3 of 8 pages
2
2
Let’s first work through all paths from ∏1/2, v∏ to remote state and back to ∏1/2, v∏.
“RAILROAD” diagrams, to keep track of second-order perturbation theory paths.
matrix element
+Bv∏ v∏′ y 2 − A v∏ v∏′ 2
same
2
∏1/2, v′∏
−Bv∏ v∏′ (y 2 −1)
1/2
2
α
∏1/2, v∏
s
′
v∏ v∑
+β
s
′
v∏ v∑
∏3/2, v′∏
same
∑s, v′∑
same
2
⎡⎣1  (–1) y⎤⎦
s
2
2
∏1/2, v∏
same
other states
[2∆3/2, 4∆3/2, 4∏3/2, 4 ∑s3/2 ]
collect terms and sum
H
⎛ e⎞
B2vv′ ( y 4 + y 2 − 1) + A 2vv′ 4 − Bvv′ A vv′ y 2
⎜⎝ f ⎟⎠ = ∑
E ov∏ − E vo∏′
v∏
′
(2)
2
∏1/2 , 2 ∏1/2
+∑
( α ) + (β )
2
s
v∏ v∑′
2
s
v∏ v∑′
v∑′
(
Now define some 2nd-order parameters.
D≡−
B2v∏ v∏′
∑
′
v∏
AD ≡ 2
≠v∏
A v∏ v∏′ Bv∏ v∏′
∑
′
v∏
E ov∏ − E ov∏′
≠v∏
′
v∏
2
A v∏ v∏′
∑
A0 ≡
(defined so that D > 0 for v∏ = 0)
E ov∏ − E ov∏′
o
o
≠v∏ E v∏ − E v∏′
(α )
o( ∑ ) ≡ ∑
E −E
2
2
s
v∏ v∑′
s
v∑′
(≠ v∑ )
p( ∑ ) ≡ 4 ∑
2
s
v∑′
(≠ v∑ )
o
v∏
o
v∑′
αsv∏ v∑′ βsv∏ v∑′
E ov∏ − E ov∑′
(β )
q( ∑ ) ≡ 2 ∑
E −E
2
2
s
v∏ v∑′
s
v∑′
(≠ v∑ )
o
v∏
o
v∑′
[HSO ⊗ HSO]
[HSO ⊗ HROT]
[HROT ⊗ HROT]
)
⎡⎣1  ( −1)s 2y + y 2 ⎤⎦ + α sv v′ βsv v′ 2 ⎣⎡1  ( −1)s y ⎤⎦
∏ ∑
∏ ∑
E ov∏ − E ov∑′
5.80 Lecture #19
Fall, 2008
Page 4 of 8 pages
Thus
H 2(2)∏
⎛ e⎞
1
4
2
2
2 s
(
=
−D
y
+
y
−
1
−
A
y
+
A
4
+
o
∑)
(
)
2
D
0
1/2 , ∏1/2 ⎜ f ⎟
2
⎝ ⎠
(no Λ-doubling)
1
1
+ p ( 2 ∑s ) ⎡⎣1  (−1)s y⎤⎦ + q ( 2 ∑s ) ⎡⎣1  (−1)s 2y + y 2 ⎤⎦
2
2
(Λ-doubling)
(Λ-doubling)
These same parameters appear in other locations in 2∏ Heff.
Non-Lecture
−Bv∏ v∏′ ( y 2 − 1)
1/ 2
2
A v∏ v′∏ 2 + Bv∏ v′∏ ( y − 2 )
same
∏1/2, v′∏
2
2
−β
s
′
v∏ v∑
∏3/2, v∏
2
(y – 1)
1/2
2
etc.
∏3/2, v′∏
same
∑s, v′∑
same
2
2
∏3/2, v∏
same
other states
Thus
H 2(2)∏
3/2 ,
2
⎛ e⎞
1
1 2 s
4
2
2
⎡
=
−D
y
−
3y
+
3⎤
+
A
y
−
2
+
A
4
+
q ( ∑ ) ⎡⎣ y 2 − 1⎤⎦
(
)
⎣
⎦
D
0
∏ 3/2 ⎜ ⎟
2
2
⎝f⎠
−Bvv′
( y 2 −1)
1/2
2
A vv′ 2 + Bvv′ ( y 2 − 2)
2
∏3/2, v∏
−β
s
v∏ v′∑
(y
2
−1)
−A vv′ 2 + Bvv′ y 2
∏1/2, v′∏
−Bvv′ ( y 2 −1)
1/2
2
∏3/2, v′∏
1/2
2
∑s, v′∑
α
s
v∏ v′∑
+β
s
v∏ v′∑
⎡⎣1  (−1)s y⎤⎦
2
∏1/2, v∏
Thus
H 2(2∏)
2
3/2 ,
⎛ e⎞
⎡ y 2 ( y 2 − 1)1/2 + ( y 2 − 2 )( y 2 − 1)1/ 2 ⎤ + 1 A ⎡ 1 ( y 2 − 1)1/2 − 1 ( y 2 − 1)1/2 ⎤
=
+D
⎣
⎦ 2 D ⎢⎣ 2
∏1/2 ⎜ ⎟
⎥⎦
2
⎝f⎠
2
2
1/2
2(y – 1)(y – 1)
0
1/2
1/2
1
1
+ p ( 2 ∑s ) ⎡⎣ − ( y 2 − 1) ⎤⎦ + q ( 2 ∑s ) ⎡⎣ − ⎡⎣1  (−1)s y⎤⎦ ( y 2 − 1) ⎤⎦
4
2
3/2
1/ 2
1/ 2
1
1
= +D2 ( y 2 − 1) − p ( y 2 − 1) − q ( 2 ∑s ) (1  (−1)s y )( y 2 − 1)
4
2
Λ-doubling
5.80 Lecture #19
Fall, 2008
Page 5 of 8 pages
Non-Lecture
Bv∑ v′∑ ( y 2  (−1)s y )
−βv∑ v∏′ ( y 2 −1)
2
1/2
2
∑sv∑
α
s
v∑ v′∏
+β
s
v∑ v′∏
(1  (–1) y)
s
2
same
∑sv′∑
same
∏3/2, v′∏
∏1/2, v′∏ same
other states same
2 –s 2
∑ , ∏1/2, 4∑–s, 4∏1/2
2
H 2(2)∑s ,2 ∑s = −D ∑ ⎡⎣ y 4  (−1)s 2y 3 + y 2 ⎤⎦
1
+ q ∑ ( 2 ∏ ) ⎡⎣ y 2 − 1 + (1  (−1)s 2y + y 2 ) ⎤⎦
2
1
+ p ∑ ( 2 ∏ ) ⎡⎣ 2 (1  (−1)s y ) ⎤⎦
4
+o ∑ ( 2 ∏ )
=
{
2
∑s, v∑
1
q ∑ ( 2 ∏ ) ⎡⎣ 2y 2  (−1)s 2y ⎤⎦
2
q∑(2∏) is exactly correlated with B∑ because it has same J-dependence.
o∑(2∑) is exactly correlated with E∑.
1
2
2p∑( ∏)
is exactly correlated with γ∑.
These second-order parameters cannot be determined by a fit to the observed energy levels. They also
cause the microscopic mechanical meaning of the E, B, γ parameters to be contaminated.
}
5.80 Lecture #19
Fall, 2008
Page 6 of 8 pages
2
2
s
eff
Now that I have worked out all of the correction terms for the ∏, ∑ H , we can examine the structure
of this matrix. For simplicity, specialize to 2∑+ (s = 0).
()
2
H(2) ef
2
∏3/2
–D∏(y4 – 3y2 + 3)
∏1/2
2
∏1/2
∑+
+D∏2(y2 – 1)3/2
1
–4p∏(y2 – 1)1/2
+2q∏(y2 – 1)
+ A0/4
1
–2q∏(1  y)(y2 – 1)1/2
sym
–D∏(y4 + y2 – 1) + A0/4
+2AD(y2 – 2)
2
2
∏3/2
1
1
1
+2ADy2 + o∏
1
1
+2p∏(1  y) + 2q∏[1  2y + y2]
2
–D∑(y4  2y3 + y2)
+q∑(y2  y)
∑+
1
+2p∑(1  y) + o∑
NOTE:
** Centrifugal Distortion matrix elements are not trivial replacement of B by [B – DJ(J + 1)]
** e/f degeneracy in 2∏ is lifted in H(2)
** all Λ-doubling in 2∏ states comes from 2∑±, none from 2∏, 2∆, 4∏, 4∆, etc.
Now apply perturbation theory to H(0) + H(1) + H(2) matrices to analyze where specific effect
(e.g. Λ-doubling) originates.
Often want to do this in order to:
* identify parameter responsible for an observed splitting with a certain J-dependence;
* prove that two fit parameters are correlated and therefore not independently determinable;
* build in correction for expected not-quite-remote perturber;
* determine whether a certain fit parameter can actually be determined by the information
contained in your specific data set.
5.80 Lecture #19
Fall, 2008
Page 7 of 8 pages
EXAMPLE - Λ-Doubling
EXPLICIT
IMPLICIT
E2∏
1/2
E2∏
3/2
e
− E2∏
e
− E2∏
1/2
3/2
e/f dependence on-diagonal in Heff
e/f dependence off-diagonal in Heff
f
= −yp ∏ − 2yq ∏ + “second order”
f
=0
“second order” =
+ “second order”
2
largest parity
largest parity
H 3/2,1/2
≈
+
o
o
E 3/2 − E1/2 dependent term independent term
largest term
H 3/ 2,1/2 = − Bv∏ ( y 2 − 1)
1/2
+ D ∏ 2 ( y 2 − 1)
3/2
−
1/2
1 /2
1
1
p ∏ ( y 2 − 1) − q ∏ (1  y ) ( y 2 − 1)
4
2
2
parity dependent part of H 3/2,1/2
1/2
1/2
1
2
H 3/2,1/2
= 2 Bv∏ q ∏ y ( y 2 − 1) ( y 2 − 1) = Bv∏ q ∏ y ( y 2 − 1)
2
o
o
E 3/2 − E1/2 ≈ A ∏
So
E 3/2e − E 3/2f
B
B 3
2
≈ −2 qy ( y − 1) ≈ −2 qJ
A
A
Similar algebra for 2∏1/2:
J
E1/2e − E1/2f ≈ − ( p ∏ + 2q ∏ ) y +



from H (2)
2
∏ 1/2 , 2 ∏ 1/2
Usually p ∏  q ∏ because p ∝ αβ
q ∝ β2
p/q≈
α A
=
β B
B 3
2 qJ

A 

from ( H 3/2,1/2 )
2
A
5.80 Lecture #19
Fall, 2008
Page 8 of 8 pages
At low-J, leading contribution to Λ-doubling
in 2∏1/2
is
–Jp∏
linear in J
2
3
in ∏3/2
is
–(2Bq/A)J cubic in J
Structure of 2∑+ state
lumped into E v∑
⎛ 2 + e⎞
1
E ⎜ ∑ ⎟ = ( E v∑ + o ∑ ) + ( Bv∑ + q ∑ ) ( y 2  y) + p ∑ (1  y) − D∑ ( y 4 2y 3 + y 2 )
f⎠
2
⎝
same as γR·S
lumped into Bv∑
A mixture of mechanical and magnetic significance is what we determine by fitting a spectrum!
Finally, replace y by N as follows:
⎛ e⎞
for 2∑+ ⎜ ⎟
⎝ f⎠
y2  y
1y
y4  2y3 + y2
e
J = N + 1/2 (F1)
y=N+1
N(N + 1)
–N
N2(N + 1)2
f
J = N – 1/2 (F2)
y=N
N(N + 1)
1+N
N2(N + 1)2
[Fi labels: for isolated 2S+1∑ state, F1 is N = J – S and lies at lowest E for given J and F2S+1 is N = J + S
and lies at highest E for given J.]
⎛ 2 + e⎞
⎤
1 ⎡1
2
E ⎜ ∑ ⎟ = E v∑ + Bv∑ N(N + 1) − D∑ [ N(N +1)] + p ∑ ⎢ (N + 1 / 2)⎥
⎦
f⎠
2 ⎣2
⎝
N is pattern-forming
quantum number!
E 2 ∑+ e − E 2 ∑+ f = −yp ∑ = − (N +1 / 2 ) p ∑
for same N (different J)